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Chirality Cannot Be Ferroic in Enantiomorphic Space-Groups

F. Gómez-Ortiz, S. Mamoudou Taganga, E. E. McCabe, A. H. Romero, E. Bousquet

Abstract

With growing interest in structural chirality in periodic solids, it has been suggested that chirality might constitute a new ferroic order parameter. In this work, we demonstrate, by means of a formal group-theoretical proof and a systematic group-subgroup analysis, that achiral-to-chiral transitions that produce either member of an enantiomorphic pair cannot be driven by a Brillouin-zone-center ($Γ$-point) instability from a common achiral parent. We further substantiate this result by explicitly showing that none of the achiral parent space groups that admit symmetry-chiral phonon eigenvectors host them at the zone center. Given that a primary ferroic order parameter must, among other requirements, transform according to a zone-center irreducible representation, we conclude that phase transitions leading to enantiomorphic space groups cannot be classified as primary ferroic transitions. This predicts that any critical enhancement of fluctuations must occur at finite-$q$ rather than as a divergence of a uniform macroscopic susceptibility.

Chirality Cannot Be Ferroic in Enantiomorphic Space-Groups

Abstract

With growing interest in structural chirality in periodic solids, it has been suggested that chirality might constitute a new ferroic order parameter. In this work, we demonstrate, by means of a formal group-theoretical proof and a systematic group-subgroup analysis, that achiral-to-chiral transitions that produce either member of an enantiomorphic pair cannot be driven by a Brillouin-zone-center (-point) instability from a common achiral parent. We further substantiate this result by explicitly showing that none of the achiral parent space groups that admit symmetry-chiral phonon eigenvectors host them at the zone center. Given that a primary ferroic order parameter must, among other requirements, transform according to a zone-center irreducible representation, we conclude that phase transitions leading to enantiomorphic space groups cannot be classified as primary ferroic transitions. This predicts that any critical enhancement of fluctuations must occur at finite- rather than as a divergence of a uniform macroscopic susceptibility.
Paper Structure (4 sections, 1 theorem, 5 equations, 1 figure, 18 tables)

This paper contains 4 sections, 1 theorem, 5 equations, 1 figure, 18 tables.

Table of Contents

  1. Supplementary Information for Chirality Cannot Be Ferroic in Enantiomorphic Space-Groups
  2. Minimum required $k$-index
  3. Chiral phonons from achiral space-groups
  4. Chiral phonons from Sohncke non-enantiomorphic space-groups

Key Result

Theorem 1

Let $L_1$ and $L_2$ be an enantiomorphic pair. Then, any space group $H$ that is a common supergroup of both enantiomorphs must satisfy $[T_H:T]>1$.

Figures (1)

  • Figure 1: Illustrative example of the theorem applied to the enantiomorphic screw axis $P4_1$ and $P4_3$.

Theorems & Definitions (2)

  • Theorem 1
  • proof